The House careened toward a Thursday vote to hold Attorney General Eric H. Holder Jr. in contempt of Congress after Republicans rejected the Justice Department’s final offer for turning over a limited set of documents, saying it wasn’t sufficient to complete their investigation into the botched gunwalking operation known as Fast and Furious.

At least a handful of Democrats have said they would vote with the GOP majority to hold Mr. Holder in contempt, raising the bipartisan peril for the White House, which has argued that the investigation is Republican-led “political theater.”

Amping up pressure on lawmakers, the National Rifle Association said it will penalize lawmakers who vote against the contempt motion in its powerful year-end scorecard — a prod some Democrats will be reluctant to ignore.

Both sides said they were reluctant to approach the vote — the first time an attorney general, the nation’s top law enforcement official, would have been cited for contempt. But their last-minute efforts to reach an agreement to head off the vote appeared to have failed, and House Speaker John A. Boehner said his troops were moving ahead.

“It’s an unfortunate place where we are. But our members raise their right hand and swear to uphold the Constitution of the United States and the laws of the United States. And we’re going to do our job,” said Mr. Boehner, Ohio Republican.

But Rep. Elijah E. Cummings, the ranking Democrat on the House Oversight and Government Reform Committee, pleaded with Mr. Boehner to avert the vote, saying the investigation has been too partisan. He identified a list of what he called 100 “errors, omissions and mischaracterizations” in the report that the committee drew up recommending the contempt citation.

“I hope after reviewing this information you will withdraw the contempt resolution, accept the attorney general’s offer to meet personally, and enter into direct negotiations with him as part of a good-faith effort to resolve this matter,” Mr. Cummings said in a letter to Mr. Boehner.

Fast and Furious was intended to track sales of U.S. guns and watch the guns be shipped across the border to a Mexican drug cartel. But the Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF) lost track of the roughly 2,000 weapons after they were sold. Some of the guns eventually began showing up at crime scenes, including two that were recovered at the site of a 2010 Arizona shootout that left Border Patrol Agent Brian Terry dead.

Mr. Terry’s family has demanded answers, which has helped push the investigation forward.

Indeed, that was one of the considerations that won over Rep. Jim Matheson, Utah Democrat, who told his hometown newspapers this week that he will vote for the contempt citation.

“It just compounds the tragedy when both sides play politics instead of releasing the facts. The Terry family, the public and Congress deserve answers,” he told the Salt Lake Tribune. “Sadly, it seems that it will take holding the attorney general in contempt to communicate that evasiveness is unacceptable.”

But Rep. Jim Cooper, Tennessee Democrat, said he would vote against contempt because the GOP was “rushing to judgment.” He urged all sides to wait for a Justice Department inspector general’s report, which has been in the works for more than a year.

The administration initially said it never knowingly let guns “walk,” but months later officially recanted that assertion and acknowledged it had done so.

Mr. Holder shut down Fast and Furious, and the Justice Department has turned over thousands of pages of documents related to the operation, but it has rejected the House oversight committee’s request for records that would shed light on Justice officials’ deliberations as they realized and, after months of delay, corrected their claim that no guns had been allowed to walk.

After months of defying a subpoena for the documents, President Obama last week claimed the documents were protected from congressional scrutiny by executive privilege, which shields White House decision-making so the president can get the best advice from his advisers.

On Wednesday, White House press secretary Jay Carney told the GOP to forgo the vote, saying lawmakers should focus instead on the economy.

“I cannot imagine this will sit well with most Americans,” he said.

Mr. Carney said senior White House and Justice Department officials met Tuesday with House Republican staffers and showed them “a representative sample” of documents sought by the committee. But they were not able to resolve the impasse.

“This was a good-faith effort to try to reach an accommodation,” Mr. Carney said.

A contempt citation likely would not have any immediate concrete effect on the standoff. Congress does have authority to arrest anyone who does not comply with a congressional demand, but that power has been “long dormant,” according to the Congressional Research Service.

Short of that, Congress must rely on either the executive branch — in this case Mr. Holder, as attorney general — to enforce the subpoena, or must turn to the courts and ask them to get involved in the clash between the other two branches of government.

The most recent contempt votes were held in 2008, when Democrats were in control and were seeking White House documents related to the firings of U.S. attorneys. Most Republicans boycotted that vote, but three of them joined Democrats as the chamber held President Bush’s top attorney and his chief of staff in contempt by a vote of 223-32.